rq: Quantile Regression

Description

Returns an object of class "rq" "rqs" or "rq.process" that represents a quantile regression fit.

Usage

rq(formula, tau=.5, data, subset, weights, na.action,
   method="br", model = TRUE, contrasts, ...)

Value

See rq.object and rq.process.object for details. Inferential matters are handled with summary. There are extractor methods logLik and AIC that are potentially relevant for model selection.

Arguments

formula

a formula object, with the response on the left of a ~ operator, and the terms, separated by + operators, on the right.

tau

the quantile(s) to be estimated, this is generally a number strictly between 0 and 1, but if specified strictly outside this range, it is presumed that the solutions for all values of tau in (0,1) are desired. In the former case an object of class "rq" is returned, in the latter, an object of class "rq.process" is returned. As of version 3.50, tau can also be a vector of values between 0 and 1; in this case an object of class "rqs" is returned containing among other things a matrix of coefficient estimates at the specified quantiles.

data

a data.frame in which to interpret the variables named in the formula, or in the subset and the weights argument. If this is missing, then the variables in the formula should be on the search list. This may also be a single number to handle some special cases -- see below for details.

subset

an optional vector specifying a subset of observations to be used in the fitting process.

weights

vector of observation weights; if supplied, the algorithm fits to minimize the sum of the weights multiplied into the absolute residuals. The length of weights must be the same as the number of observations. The weights must be nonnegative and it is strongly recommended that they be strictly positive, since zero weights are ambiguous.

na.action

a function to filter missing data. This is applied to the model.frame after any subset argument has been used. The default (with na.fail) is to create an error if any missing values are found. A possible alternative is na.omit, which deletes observations that contain one or more missing values.

model

if TRUE then the model frame is returned. This is essential if one wants to call summary subsequently.

method

the algorithmic method used to compute the fit. There are several options:

"br" The default method is the modified version of the Barrodale and Roberts algorithm for $l_{1}$ -regression, used by l1fit in S, and is described in detail in Koenker and d'Orey(1987, 1994), default = "br". This is quite efficient for problems up to several thousand observations, and may be used to compute the full quantile regression process. It also implements a scheme for computing confidence intervals for the estimated parameters, based on inversion of a rank test described in Koenker(1994).
"fn" For larger problems it is advantageous to use the Frisch--Newton interior point method "fn". This is described in detail in Portnoy and Koenker(1997).
"pfn" For even larger problems one can use the Frisch--Newton approach after preprocessing "pfn". Also described in detail in Portnoy and Koenker(1997), this method is primarily well-suited for large n, small p problems, that is when the parametric dimension of the model is modest.
"sfn" For large problems with large parametric dimension it is often advantageous to use method "sfn" which also uses the Frisch-Newton algorithm, but exploits sparse algebra to compute iterates. This is especially helpful when the model includes factor variables that, when expanded, generate design matrices that are very sparse. At present options for inference, i.e. summary methods are somewhat limited when using the "sfn" method. Only the option se = "nid" is currently available, but I hope to implement some bootstrap options in the near future.
"fnc" Another option enables the user to specify linear inequality constraints on the fitted coefficients; in this case one needs to specify the matrix R and the vector r representing the constraints in the form $R b \geq r$ . See the examples below.
"conquer" For very large problems especially those with large parametric dimension, this option provides a link to the conquer of He, Pan, Tan, and Zhou (2020). Calls to summary when the fitted object is computed with this option invoke the multiplier bootstrap percentile method of the conquer package and can be considerably quicker than other options when the problem size is large. Further options for this fitting method are described in the conquer package. Note that this option employs a smoothing form of the usual QR objective function so solutions may be expected to differ somewhat from those produced with the other options.
"pfnb" This option is intended for applications with large sample sizes and/or moderately fine tau grids. It uses a form of preprocessing to accelerate the solution process. The loop over taus occurs inside the Fortran call and there should be more efficient than other methods in large problems.
"qfnb" This option is like the preceeding one except that it doesn't use the preprocessing option.
"ppro" This option is an R prototype of the pfnb and is offered for historical/interpretative purposes, but probably should be considered deprecated.
"lasso" There are two penalized methods: "lasso" and "scad" that implement the lasso penalty and Fan and Li smoothly clipped absolute deviation penalty, respectively. These methods should probably be regarded as experimental. Note: weights are ignored when the method is penalized.

contrasts

a list giving contrasts for some or all of the factors default = NULL appearing in the model formula. The elements of the list should have the same name as the variable and should be either a contrast matrix (specifically, any full-rank matrix with as many rows as there are levels in the factor), or else a function to compute such a matrix given the number of levels.

...

additional arguments for the fitting routines (see rq.fit.br and rq.fit.fnb, etc. and the functions they call).

Method

The function computes an estimate on the tau-th conditional quantile function of the response, given the covariates, as specified by the formula argument. Like lm(), the function presumes a linear specification for the quantile regression model, i.e. that the formula defines a model that is linear in parameters. For non-linear (in parameters) quantile regression see the package nlrq(). The function minimizes a weighted sum of absolute residuals that can be formulated as a linear programming problem. As noted above, there are several different algorithms that can be chosen depending on problem size and other characteristics. For moderate sized problems ( $n ≪ 5, 000, p ≪ 20$ ) it is recommended that the default "br" method be used. There are several choices of methods for computing confidence intervals and associated test statistics. See the documentation for summary.rq for further details and options.

Details

For further details see the vignette available from R with vignette("rq",package="quantreg") and/or the Koenker (2005). For estimation of nonlinear (in parameters) quantile regression models there is the function nlrq and for nonparametric additive quantile regression there is the function rqss. Fitting of quantile regression models with censored data is handled by the crq function.

References

[1] Koenker, R. W. and Bassett, G. W. (1978). Regression quantiles, Econometrica, 46, 33--50.

[2] Koenker, R.W. and d'Orey (1987, 1994). Computing regression quantiles. Applied Statistics, 36, 383--393, and 43, 410--414.

[3] Gutenbrunner, C. Jureckova, J. (1991). Regression quantile and regression rank score process in the linear model and derived statistics, Annals of Statistics, 20, 305--330.

[4] Xuming He and Xiaoou Pan and Kean Ming Tan and Wen-Xin Zhou, (2020) conquer: Convolution-Type Smoothed Quantile Regression, https://CRAN.R-project.org/package=conquer

[4] Koenker, R. W. (1994). Confidence Intervals for regression quantiles, in P. Mandl and M. Huskova (eds.), Asymptotic Statistics, 349--359, Springer-Verlag, New York.

[5] Koenker, R. and S. Portnoy (1997) The Gaussian Hare and the Laplacean Tortoise: Computability of Squared-error vs Absolute Error Estimators, (with discussion). Statistical Science, 12, 279-300.

[6] Koenker, R. W. (2005). Quantile Regression, Cambridge U. Press.

There is also recent information available at the URL: http://www.econ.uiuc.edu/~roger/.

Examples

Run this code

data(stackloss)
rq(stack.loss ~ stack.x,.5)  #median (l1) regression  fit for the stackloss data. 
rq(stack.loss ~ stack.x,.25)  #the 1st quartile, 
        #note that 8 of the 21 points lie exactly on this plane in 4-space! 
rq(stack.loss ~ stack.x, tau=-1)   #this returns the full rq process
rq(rnorm(50) ~ 1, ci=FALSE)    #ordinary sample median --no rank inversion ci
rq(rnorm(50) ~ 1, weights=runif(50),ci=FALSE)  #weighted sample median 
#plot of engel data and some rq lines see KB(1982) for references to data
data(engel)
attach(engel)
plot(income,foodexp,xlab="Household Income",ylab="Food Expenditure",type = "n", cex=.5)
points(income,foodexp,cex=.5,col="blue")
taus <- c(.05,.1,.25,.75,.9,.95)
xx <- seq(min(income),max(income),100)
f <- coef(rq((foodexp)~(income),tau=taus))
yy <- cbind(1,xx)%*%f
for(i in 1:length(taus)){
        lines(xx,yy[,i],col = "gray")
        }
abline(lm(foodexp ~ income),col="red",lty = 2)
abline(rq(foodexp ~ income), col="blue")
legend(3000,500,c("mean (LSE) fit", "median (LAE) fit"),
	col = c("red","blue"),lty = c(2,1))
#Example of plotting of coefficients and their confidence bands
plot(summary(rq(foodexp~income,tau = 1:49/50,data=engel)))
#Example to illustrate inequality constrained fitting
n <- 100
p <- 5
X <- matrix(rnorm(n*p),n,p)
y <- .95*apply(X,1,sum)+rnorm(n)
#constrain slope coefficients to lie between zero and one
R <- cbind(0,rbind(diag(p),-diag(p)))
r <- c(rep(0,p),-rep(1,p))
rq(y~X,R=R,r=r,method="fnc")

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